AN EWMA CONTROL CHART FOR THE EXPONENTIAL DISTRIBUTION USING REPETITIVE SAMPLING

No. 2 2017 DOI: 10.5277/ord170201

Muhammad AZAM¹ Muhammad ASLAM² Chi-Hyuck JUN³

AN EWMA CONTROL CHART FOR THE EXPONENTIAL DISTRIBUTION USING REPETITIVE SAMPLING

A new EWMA control chart has been proposed under repetitive sampling when a quantitative characteristic follows the exponential distribution. The properties of the proposed chart, including the average run lengths has been is compared with two existing control charts with the help of simulated data. An application of the proposed chart hs been illustrated using a healthcare data set.

Keywords: control chart, EWMA statistic, exponential distribution, normal distribution, average run length

1. Introduction

Customers demand high quality products on the market. To meet customers’ re- quirements and maintain a good reputation, a producer must pay full attention to the production process from raw materials to the final product. To achieve the goal of pro- ducing a high quality product, the producer has to rely on statistical tools such as control charts and acceptance sampling plans. The former are used for monitoring the manufac- turing process and the latter are used for inspection, from the raw materials to the final

_________________________

1Department of Statistics and Computer Sciences, University of Veterinary and Animal Sciences, Lahore 54000, Pakistan, e-mail address: mazam72@yahoo.com

2Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21551, Saudi Arabia, e-mail address: aslam_ravian@hotmail.com

3Department of Industrial and Management Engineering, POSTECH, Pohang 790-784, Republic of Korea, e-mail address: chjun@postech.ac.kr

product. Control charts are powerful tools for minimizing the proportion of non-con- forming products, as they provide a quick indication of when the process is shifting to an out-of-control state. Such quick indications about shifts in a process help engineers to bring it back into an under-control state.

Shewhart control charts are simple to apply in industry. The major flaw of the Shewhart control chart is that it does not detect small changes in the process. To over- come this problem, researchers have been attempting to introduce various control charts that can detect small changes in a manufacturing process. Roberts introduced an expo- nentially weighted moving average (EWMA) control chart to detect a small shift in a process [1]. The EWMA statistic in a control chart utilizes current and past infor- mation to make a decision about the manufacturing process. The weight of past infor- mation is controlled using the smoothing constant. More details about the application and design of EWMA control charts have been published by Lucas and Saucci [2]. More information can be found in [3–6].

Mostly, control charts are designed by assuming that the quantitative characteristic under study follows the normal distribution. Such control charts are not helpful in de- tecting a shift in the process when the characteristic of interest follows some non-normal distribution. In this situation, the likelihood of false alarms may increase and/or the pro- portion of non-conforming products may increase. Furthermore, there are many situa- tions where data are not collected in subgroups. As mentioned by [7], the data not col- lected in a subgroup may follow an exponential distribution. Characteristics such as the lifetime of an item or waiting time of a customer may follow an exponential distribution.

A control chart for the exponential distribution is called a t-chart. A detailed study of the design and application of a t-chart can be read in [8].

Hence, t-charts or EWMA t-charts are designed for single sampling. Aslam et al.

designed an EWMA t-chart under a single sampling scheme [9]. The use of repetitive group sampling (RGS) in the area of control charts has attracted researchers, due to its simplicity and efficiency compared to sequential or multiple sampling schemes. RGS was originally designed by [10] and its efficiency is well verified compared to single and double sampling schemes in acceptance sampling plans. Recently, some authors designed control charts using an RGS scheme [11–13].

By exploring the literature, it can be noted that results on EWMA control charts using a single sampling scheme are available in the literature. Also, some results are available on control charts using RGS assuming that the quantity of interest follows the normal distribution. According to the best of the authors’ knowledge, there is no work available on EWMA t-charts using an RGS scheme. In this paper, we will focus on the design of a t-chart using the EWMA statistic under an RGS scheme. The proposed con- trol chart will be an extension of [9]. The complete structure of the proposed chart will be given. The efficiency of the proposed chart will be compared with [9] using simulated data. An application of the proposed chart in the health service is also given.

2. Design of the proposed control chart

It is assumed that the time between events T follows the exponential distribution with scale (mean) parameter  whose probability density function (pdf) is given by

 

f t ^ t



   (1)

If T follows the exponential distribution with mean , then T^*T^(1/3.6) follows the Weibull distribution with the shape parameter  and scale parameter ^1/ [7]. Nelson suggested that when  = 3.6, the above Weibull distribution becomes approximately a normal distribution [8].

The proposed control chart can be described as follows, as an extension of the con- trol chart proposed by [9]. It should be noted that there are two pairs of control limits in the proposed control chart.

Step 1. Select an item at random and measure its quantitative characteristic T and calculate the following transformed variable, T^*:

* (1/3.6)

T  T

Then, calculate the following EWMA statistic, denoted ,Z using the smoothing _t constant 

 

*

1 1

t t

Z T   Z_

Step 2. Declare the process as out-of-control if Z_t UCL₁ orZ_t LCL₁. Declare the process as under-control if LCL₂Z_t UCL₂. Otherwise, go to Step 1 and repeat the process.

As stated in Step 2, the proposed control chart does not make any conclusion on the state of the process if the statistic lies between the inner and outer control limits, in which case repetitive sampling is required.

Suppose first that the process is under control with specified mean 0. Because the transformed variable T^*follows an approximately normal distribution, so does the EWMA statistic. The mean _Z and the variance _Z² of the Z statistic are obtained by _t

* 0

1 1 3.6

Z   ^  ^

 

 

2 *2

0

2 1

1 1 1 1

2 3.6 3.6

t Z

     



     

           

where ₀^* ₀^1/3.6.

Therefore, the two pairs of control limits are given below. First, the outer control limits are

 

 

*

1( ) 1 0

2 2 1

1 1 3.6

2 1

1 1 1 1

2 3.6 3.6

t Z Z

t

LCL k

k

   

   



 

     

 

    

         



  





  (2)

 

 

*

1( ) 1 0

2 2 1

1 1 3.6

2 1

1 1 1 1

2 3.6 3.6

t Z Z

t

UCL k

k

   

   



 

     

 

    

         



  





  (3)

The inner control limits are

 

 

*

2( ) 2 0

2 2 2

1 1 3.6

2 1

1 1 1 1

2 3.6 3.6

t Z Z

t

LCL k

k

   

   



 

    

    

         

 



    

(4)

 

 

*

2( ) 2 0

2 2 2

1 1 3.6

2 1

1 1 1 1

2 3.6 3.6

t Z Z

t

UCL k

k

   

   



 

     

    

         



  





 

(5)

where the control constants k1 and k2 are to be determined.

Let

 

 

1 1

1 2 1

c 1 1 1 1 1

3.6 2 3.6 3.6

t

L  k    



     

            

 

 

1 1

1 2 1

1 1 1 1 1

3.6 2 3.6 3.6

t

cU  k    



     

            

 

 

2 2

1 2 1

1 1 1 1 1

3.6 2 3.6 3.6

t

cL  k    



     

            

 

 

2 2

1 2 1

1 1 1 1 1

3.6 2 3.6 3.6

t

cU  k    



     

            

then the above control limits can be rewritten as

*

1 0 L1

LCL  c

*

1 0 U1

LCU  c

*

2 0 L2

LCL  c

*

2 0 U2

LCU  c

Therefore, the probability of being declared out-of-control

 

given as follows:

0 out, 1 0

out 0

1 rep

P P

 P

 (6)

The average run length (ARL) when   ₀ is given as follows:

0 0

out

ARL 1

 P (7)

The average sample size (ASS0) for the process when   is given by ₀

0 0

rep

1 ASS 1

 P

 (8)

Suppose now that the process mean is shifted to ₁c₀, where c is a constant. The probability that the process is inferred to be out-of-control based on a single sample

1 out,1

(P ) when   ₁.

Finally, the probability that the process is declared to be out-of-control (P_out,1¹ ) and the out-of-control ARL (ARL1)when   ₁ are given as follows:

1 out, 1 1

out 1

1 rep

P P

 P

 (9)

and

1 1

out

ARL 1

P (10)

Algorithm 1. Monte Carlo simulation of an EWMA control chart for the exponential distribution using a repetitive sampling scheme when the process is under-control (c = 1).

The following are the steps involved in an R program defining the Monte Carlo simulation:

1. Computation of the proposed EWMA statistic, .Z _t

1.1. Specify the value of the required under-control ARL, denoted r and ₀ . 1.2. Generate T a random sample of size 1 for each subgroup, from the exponen- tial distribution with specified parameter ₀, i.e. under the assumption that the process is under-control (c = 1). Generate 1000 such subgroups.

1.3. Transform T into T = T^*1/ and substitute this into the EWMA statistic.

1.4. Compute the EWMA statistic, .Z _t 2. Computation of the variable control limits.

2.1. Define appropriate values of the control coefficients, k and ₁ k ₂.

2.2. Calculate LCL_{1( )t} , LCL_{2( )t} , UCL_{2( )t} and UCL_{1( )t} based on 1000 subgroups.

2.3. Keeping in view the operational strategy of the proposed control chart, infer whether the process should be declared as under-control, in repeat mode, or out- of-control. If the process is declared as in-control, repeat Steps 1.1–2.3. If the process is in repeat mode, count the number of repetitions. Otherwise, define the run length to be the number of subgroups, together with the number of repetitions, i.e. the time for which the process was declared to be either under-control or in repeat mode before being declared to be out-of-control.

3. Computation of the average run length (ARL).

3.1. Repeat Steps 1.1–2.3 a sufficient number of times (10 000 say) to calculate the under-control ARL. If the under-control ARL is equal to the specified ARL ₀, then stop the process and go to Step 3.2. Otherwise, modify the values of the control coefficients and repeat Steps 1.1–3.1.

3.2. Determine k and ₁ k such that ₂ ARL₀r₀.

4. Computation of the average run length (ARL for the shifted process. ₁) 4.1. Repeat Steps 1–3 for the values of repeat Steps 1.1–3.1.

4.2. Determine obtained in Step 3.2.

Algorithm 2. Monte Carlo simulation of an EWMA control chart for the exponen- tial distribution using a repetitive sampling scheme for the shifted process(c 1).

The following are the steps involved in an R program defining the Monte Carlo simulation:

1. Computation of the proposed EWMA statistic, .Z _t

1.1. Specify the value of the smoothing constant  and shift c.

1.2. Generate T, a random sample of size 1 for each subgroup, from the exponen- tial distribution with specified parameter ₁c₀ for a shifted process (c1).

Generate 1000 such subgroups.

1.3. Transform T into T^* T^1/ and substitute this into the EWMA statistic.

1.4. Compute the EWMA statistic, .Z _t 2. Computation of the variable control limits.

2.1. Specify the values of control coefficients k and ₁ k to be those obtained in ₂ Algorithm 1.

2.2. Calculate LCL_{1( )t} , LCL_{2( )t} , UCL_{2( )t} andUCL_{1( )t} from 1000 subgroups.

2.3. Keeping in view the operational strategy of the proposed control chart, infer whether the process should be declared as under-control, in repeat mode or out-

-of-control. If the process is declared as under-control, repeat Steps 1.1–2.3. If the process is inferred to be in repeat mode, count the number of repetitions. Oth- erwise, define the run length to be the number of subgroups, together with the number of repetitions, i.e. the length of time for which the process was declared to be under-control or in repeat mode before it is declared to be out-of-control.

3. Computation of the average run length (ARL for a shifted process. ₁)

3.1. Repeat steps 1.1–2.3 a sufficient number of times (10 000 say) to calculate the ARL for a given shifted process. ₁

The values of ARL1 and SDRL for various , ARL0, and f are given in Tables 1–3.

Table 1. ARLs and SDRLs for the proposed chart when ARL0  200

f

 = 0.2 k1 = 2.28, k2= 0.40

 = 0.4 k1= 2.24, k2= 0.33

 = 0.6 k1= 2.22, k2= 0.27

 = 0.8 k1= 2.165, k2= 0.24

 = 1.0 k1 = 2.11, k2 = 0.20

ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL

1.00 203.28 247.94 200.40 207.72 204.74 214.03 201.75 219.38 201.14 226.91 0.90 80.94 128.47 117.03 130.98 129.46 145.89 134.34 154.51 137.40 168.18 0.80 24.92 65.21 53.48 71.44 66.59 86.37 75.03 96.86 80.01 110.21 0.70 6.29 25.02 23.29 39.48 31.24 48.79 37.79 58.19 42.42 65.24 0.60 1.63 6.62 9.55 19.43 14.82 25.74 17.90 29.84 20.94 36.27 0.50 1.02 0.61 3.84 8.12 6.51 12.48 8.32 14.07 10.31 18.13

0.40 1.00 0.00 1.86 3.62 3.10 5.38 4.12 6.66 4.97 8.62

0.30 1.00 0.00 1.17 1.20 1.72 2.42 2.22 3.07 2.63 3.72

0.20 1.00 0.00 1.02 0.28 1.19 0.93 1.39 1.37 1.55 1.61

0.10 1.00 0.00 1.00 0.00 1.02 0.27 1.07 0.47 1.13 0.65

Table 2. The ARLs and SDRLs for the proposed chart when ARL0  300

f

 = 0.2 k1 = 2.59, k2 = 0.70

 = 0.4 k1 = 2.52, k2 = 0.50

 = 0.6 k1 = 2.45, k2 = 0.40

 = 0.8 k1 = 2.398, k2 = 0.36

 = 1.0 k1 = 2.30, k2 = 0.25

ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL

1.00 303.74 331.66 304.92 309.08 301.57 302.88 302.02 301.57 303.97 314.10 0.90 105.30 143.68 152.60 160.17 166.85 170.87 174.56 179.52 194.27 214.03 0.80 31.22 61.03 65.35 76.00 77.32 86.72 85.87 98.11 110.34 133.51 0.70 8.11 24.15 29.17 42.30 37.79 50.04 43.20 56.85 55.28 77.86 0.60 2.24 8.68 12.66 23.66 18.15 29.53 21.81 32.57 26.43 42.74 0.50 1.05 1.10 5.15 11.28 7.97 14.64 10.11 16.94 12.59 22.01

0.40 1.00 0.00 2.10 3.84 3.71 6.76 5.00 8.16 5.63 9.40

0.30 1.00 0.00 1.26 1.57 1.88 2.76 2.46 3.60 2.89 4.06

0.20 1.00 0.00 1.02 0.31 1.21 1.01 1.47 1.48 1.67 2.00

0.10 1.00 0.00 1.00 0.00 1.02 0.23 1.08 0.51 1.15 0.72

Table 3. The ARLs and SDRLs for the proposed chart when ARL0  370

f

 = 0.2 k1 = 2.70, k2 = 0.92

 = 0.4 k1 = 2.75, k2 = 0.86

 = 0.6 k1 = 2.70, k2 = 0.81

 = 0.8 k1 = 2.64, k2 = 0.75

 = 1.0 k1 = 2.64, k2 = 0.75

ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL

1.00 371.08 369.66 370.59 342.83 371.49 346.79 374.70 354.12 371.39 356.32 0.90 138.65 175.32 170.79 175.69 179.12 179.04 182.01 182.72 189.95 188.53 0.80 38.10 65.13 62.97 65.57 71.03 72.33 77.14 75.89 86.42 86.02 0.70 10.70 28.95 27.59 33.58 32.69 35.40 36.47 38.06 42.35 44.70 0.60 2.83 10.18 12.78 20.42 16.21 20.92 18.58 23.17 21.23 24.24 0.50 1.16 3.51 5.59 10.52 7.85 11.52 9.59 13.37 11.34 15.03

0.40 1.00 0.18 2.43 4.54 3.86 6.30 5.00 7.40 6.13 8.74

0.30 1.00 0.00 1.31 1.59 2.02 2.84 2.51 3.35 3.11 4.12

0.20 1.00 0.00 1.03 0.38 1.25 1.06 1.51 1.48 1.75 1.94

0.10 1.00 0.00 1.00 0.00 1.03 0.30 1.09 0.52 1.16 0.72

3. Comparative study

In this section, we will discuss the advantages of the proposed chart over the existing t-charts proposed in [14] and [7]. The proposed chart becomes the one in [14] when  = 1 and the chart from [7] when k₁k₂ andk  = 1. To compare the performance of the proposed control chart with the existing control charts, simulation will be used. Assume that r0 = 370 and  = 0.20. The data are generated from the exponential distribution with

0 = 0.21. The first 25 samples are generated from the in control process and the next 25 samples are generated from the shifted process with c 1.4. Statistics describing T^* and Mt for these three control charts are reported in Table 4.

Fig. 1. Proposed control chart for simulated data 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 6 11 16 21 26 31 36 41 46

Zt

Sample number

data LCL1 LCL2 UCL2 UCL1

Table 4. Simulated data Proposed chart

 = 0.20

Chart from [7]

 = 1.0

Chart from [9]

 = 0.20

Proposed chart

 = 0.20

Chart from [7]

 = 1.0

Chart from [9]

 = 0.20

T^* Zt T^* Zt T^* Zt T^* Zt T^* Zt T^* Zt

0.4402 0.5063 0.7267 0.7267 0.5447 0.5644 0.6523 0.5181 0.6671 0.6671 0.4459 0.5276 0.3601 0.4771 0.4463 0.4463 0.8661 0.6248 0.4073 0.4960 0.7442 0.7442 0.5075 0.5236 0.5680 0.4952 0.6807 0.6807 0.3026 0.5603 0.5823 0.5132 0.4952 0.4952 0.2149 0.4618 0.4962 0.4954 0.5104 0.5104 0.3848 0.5252 0.8030 0.5712 0.5969 0.5969 0.542 0.4779 0.6687 0.5301 0.3183 0.3183 0.5498 0.5301 0.5966 0.5763 0.7707 0.7707 0.4651 0.4753 0.5634 0.5367 0.5813 0.5813 0.5598 0.5361 0.3888 0.5388 0.4489 0.4489 0.5512 0.4905 0.4517 0.5197 0.5744 0.5744 0.4293 0.5147 0.5208 0.5352 0.6623 0.6623 0.7106 0.5345 0.7850 0.5728 0.6853 0.6853 0.4303 0.4978 0.4253 0.5132 0.5257 0.5257 0.5371 0.535 0.2637 0.5110 0.6210 0.6210 0.4644 0.4911 0.3657 0.4837 0.4737 0.4737 0.6148 0.551 0.5177 0.5123 0.4989 0.4989 0.539 0.5007 0.6728 0.5215 0.5361 0.5361 0.7189 0.5846 0.2576 0.4614 0.5438 0.5438 0.6304 0.5266 0.9897 0.6152 0.5481 0.5481 0.5929 0.5862 0.5444 0.4780 0.7663 0.7663 0.6786 0.557 0.3812 0.5684 0.6275 0.6275 0.2852 0.526 0.3854 0.4594 0.3178 0.3178 0.5505 0.5557 0.4639 0.5475 0.6604 0.6604 0.6003 0.5409 0.9001 0.5476 0.2777 0.2777 0.7011 0.5848 0.7901 0.5960 0.3968 0.3968 0.8309 0.5989 0.4921 0.5365 0.7989 0.7989 0.7093 0.6097 0.4850 0.5738 0.6867 0.6867 0.6885 0.6168 0.6303 0.5552 0.6044 0.6044 0.5669 0.6012 0.7543 0.6099 0.6739 0.6739 0.5489 0.6032 0.5709 0.5584 0.6400 0.6400 0.4906 0.579 0.4082 0.5696 0.8083 0.8083 0.5388 0.5903 0.5159 0.5499 0.3879 0.3879 0.8931 0.6419 0.9154 0.6387 0.2794 0.2794 0.5555 0.5834 0.5252 0.5449 0.5120 0.5120 0.5367 0.6208 0.6583 0.6427 0.5283 0.5283 0.1884 0.5044 0.7641 0.5888 0.6210 0.6210 0.8278 0.6622 0.1612 0.5464 0.7662 0.7662 0.4517 0.4939 0.6121 0.5934 1.0118 1.0118 0.3394 0.5977 0.7501 0.5871 0.9588 0.9588 0.6349 0.5221 0.6479 0.6043 0.6051 0.6051 0.7126 0.6206 0.8957 0.6488 0.6016 0.6016 0.5402 0.5257 0.5071 0.5849 0.6044 0.6044 0.5848 0.6135 0.7774 0.6746 0.5151 0.5151 0.4395 0.5085 0.2580 0.5195 0.4259 0.4259 0.8198 0.6547 0.8131 0.7023 0.7866 0.7866 0.1604 0.4388 0.3450 0.4846 0.8123 0.8123 0.1211 0.548 0.3192 0.6257 0.7745 0.7745 0.5426 0.4596

The four control limits of the proposed chart with k₁3.09 and k₂1.10 are given as follows: LCL₁0.3984,UCL₁0.7697, LCL₂ 0.5180, and UCL₂0.6502.

The proposed control chart is given in Fig. 1. It can be seen that the 46th sample is declared to be out of control, while the tabulated value of ARL is about 22. Hence, the ₁ proposed control has the ability to detect shifts in the process as shown in Fig. 1. Now we see whether the existing control chart proposed by [14] detects shifts or not. The four control limits of the chart given by [9] with k₁3.09 and k₂ 1.10 are given as follows: LCL₁0.0272, UCL₁1.1410, LCL₂ 0.3858and UCL₂0.7824.

The data are also plotted for the chart proposed by [11] in Fig. 2. It can be seen that all the values of Mt are within the control limits. The chart proposed in [11] shows that some values of the plotted statistic are within the repetition area, but did not indicate a shift in the process. Similarly, to prove the increased efficiency of the proposed control

chart compared to the chart proposed by [7], the values of the statistic are plotted in Fig. 3 with k3.0, LCL0.4039and UCL0.7643. From Figure 3, it can be seen that all the values of the simulated data are within the control limits. This chart shows no shift in the manufacturing process.

Fig. 2. Chart for simulated data based on the procedure from [9]

Fig. 3. Chart for simulated data based on the procedure from [7]

Hence, from Figures 1–3, it can be seen that the proposed control chart performs better than the two existing control charts in providing a quick indication of a shift in the process. Thus appropriate introduction of the proposed control chart will lead to a fall in the frequency of faulty goods. It has been noted that the chart proposed by [11]

0.0 0.2 0.4 0.6 0.8 1.0

1 6 11 16 21 26 31 36 41 46

Zt

Sample number

data LCL1 LCL2 UCL2 UCL1

0 0.2 0.4 0.6 0.8 1 1.2

1 6 11 16 21 26 31 36 41 46

Zt

Sample number

data LCL1 UCL1

is better than the one proposed in [7], so we will only compare the proposed control chart with the one from [11].

Fig. 4. Plots of ARLs for various values of  when ARL0  200

Fig. 5. Plots of ARLs for various values of  when ARL0  300

For this purpose, we compare the ARLs for various specified parameters, illustrated in Figs. 4–6. It can be seen from these curves that, in each case, the proposed control chart gives smaller values of the ARLs as compared to the chart defined in [11]. For example, from Fig. 4, it can be seen that the curve corresponding to the proposed chart is below the curves corresponding to the two other control charts, which shows that the proposed chart is better than the existing charts.

0 50 100 150 200 250

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

ARLs

Shift

lambda = 0.2 lambda=0.4 lambda=0.6 lambda=0.8 lambda=1.0

0 50 100 150 200 250 300 350

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

ARLs

Shift

lambda=0.2 lambda=0.4 lamda=0.6 lambda=0.8 lambda=1.0

Fig. 6. Plots of ARLs for various values of  when ARL0  370

4. Industrial application

In this section, we will present the application of the proposed chart in healthcare by using data regarding urinary tract infections (UTIs). The same data were used by [7] and [9]. The use of control charts for monitoring long-term health are described in [15] and [16].

The data are taken from [7]. Let r0 = 370. The values of the statistic Mt for the proposed control chart are plotted in Fig. 7. Note that several points UCL and some points are below ₂

2,

LCL while the control chart defined in [7] shows that the process is under control.

Fig. 7. Proposed chart for data describing a urinary tract infection 0

50 100 150 200 250 300 350 400

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

ARLs

Shift

lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 lambda = 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53

Value of Zt

Sample number

data LCL1 LCL2 UCL2 UCL1

5. Concluding remarks

The proposed EWMA chart is designed to utilize repetitive sampling when the quantitative characteristic of interest follows the exponential distribution. The ARLs of the proposed chart are derived for under-control and out-of-control processes. The ad- vantages of the proposed chart over two existing control charts are described. The pro- posed chart performs better than these existing charts in terms of ARLs, i.e. it provides swift indication about shifts in a process. An application of the proposed chart is illus- trated with the help of real healthcare data for monitoring UTIs. The ability of the pro- posed chart to detect shifts in a process is also compared with other existing control charts on the basis of simulated data.

Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of this manuscript. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, acknowledges with thanks the technical and financial support of the DSR.

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Received 7 December 2016 Accepted 3 May 2017